DE-BUNDESWETTBEWERB 2007

Consider a regular polygon with 2007 vertices. The natural numbers $1,2, \ldots, 4014$ shall be distributed across the vertices and edge midpoints such that for each side the sum of its three numbers (two end points, one side center) has the same value. Show that such an assignment is possible.

In triangle $ABC$ points $E$ and $F$ lie on sides $AC$ and $BC$ such that segments $AE$ and $BF$ have equal length, and circles formed by $A,C,F$ and by $B,C,E,$ respectively, intersect at point $C$ and another point $D.$ Prove that that the line $CD$ bisects $\angle ACB.$

Let $a$ be a positive integer.

How many non-negative integer solutions x does the equation

$\lfloor \frac{x}{a}\rfloor = \lfloor \frac{x}{a+1}\rfloor$

have?

$\lfloor ~ \rfloor$ ---> Floor Function.

For which numbers $n$ is there a positive integer $k$ with the following property: The sum of digits for $k$ is $n$ and the number $k^2$ has sum of digits $n^2.$

At the start of the game there are $r$ red and $g$ green pieces/stones on the table. Hojoo and Kestutis make moves in turn. Hojoo starts. The person due to make a move, chooses a colour and removes $k$ pieces of this colour. The number $k$ has to be a divisor of the current number of stones of the other colour. The person removing the last piece wins. Who can force the victory?

A set $E$ of points in the 3D space let $L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $E.$ Let $T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $L(L(T))?$